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The spatially nonlocal response functions of graphene obtained on the basisof first principles of quantum field theory using the polarization tensor areconsidered in the areas of both the on-the-mass-shell and off-the-mass-shellwaves. It s shown that at zero frequency the longitudinal permittivity ofgraphene is the regular function, whereas the transverse one possesses a doublepole for any nonzero wave vector. According to our results, both thelongitudinal and transverse permittivities satisfy the dispersion(Kramers-Kronig) relations connecting their real and imaginary parts, as wellas expressing each of these permittivities along the imaginary frequency axisvia its imaginary part. For the transverse permittivity, the form of anadditional term arising in the dispersion relations due to the presence of adouble pole is found. The form of dispersion relations is unaffected by thebranch points which arise on the real frequency axis in the presence of spatialnonlocality. The obtained results are discussed in connection with the wellknown problem of the Lifshitz theory which was found to be in conflict with themeasurement data when using the much studied response function of metals. Apossible way of attack on this problem based on the case of graphene issuggested.

Quantum field theoretical framework for the electromagnetic response of graphene and dispersion relations with implications to the Casimir effect